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An Improved Deterministic #SAT Algorithm for Small de Morgan Formulas

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Abstract

We give a deterministic #SAT algorithm for de Morgan formulas of size up to \(n^{2.63}\), which runs in time \(2^{n-n^{{\varOmega }(1)}}\). This improves upon the deterministic #SAT algorithm of Chen et al. (Proceedings of the twenty-ninth annual IEEE conference on computational complexity, 2014), which has similar running time but works only for formulas of size less than \(n^{2.5}\). Our new algorithm is based on the shrinkage of de Morgan formulas under random restrictions, shown by Paterson and Zwick (Random Struct Algorithms 4(2):135–150, 1993). We prove a concentrated and constructive version of their shrinkage result. Namely, we give a deterministic polynomial-time algorithm that selects variables in a given de Morgan formula so that, with high probability over the random assignments to the chosen variables, the original formula shrinks in size, when simplified using a given deterministic polynomial-time formula-simplification algorithm.

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Notes

  1. Constant-depth, unbounded fanin circuits, using AND, OR, NOT, and (MOD m) gates, for any integer m.

  2. Constant-depth, unbounded fanin circuits, using AND, OR, and NOT gates.

  3. Then we could let \(F_z := (x\wedge y) \vee (\overline{x} \wedge \mathbf{Simplify}( F_z|_{x=0} ))\).

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Acknowledgments

The authors would like to thank the anonymous referees for careful reading and valuable comments, which significantly improved the presentation of the paper.

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  1. Ruiwen Chen

    Present address: School of Informatics, University of Edinburgh, Edinburgh, UK

Authors and Affiliations

  1. School of Computing Science, Simon Fraser University, Burnaby, BC, Canada

    Ruiwen Chen & Valentine Kabanets

  2. Institute of Mathematical Sciences, Chennai, India

    Nitin Saurabh

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  1. Ruiwen Chen
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  2. Valentine Kabanets
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  3. Nitin Saurabh
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Correspondence to Nitin Saurabh.

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Chen, R., Kabanets, V. & Saurabh, N. An Improved Deterministic #SAT Algorithm for Small de Morgan Formulas. Algorithmica 76, 68–87 (2016). https://doi.org/10.1007/s00453-015-0020-z

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